Originally Posted by

**innuenn** 2.) If a,b is in the emement of natural numbers and both a and b are odd, then x^2+ax+b can not be factored into a product of two

linear factors (a linear factor has the form cx+d, where c,d are elements of natural numbers)

Suppose that $\displaystyle x^2+a.x+b$ with $\displaystyle a$ and $\displaystyle b$

odd, can be factored into a product of two

linear factors $\displaystyle c.x+d$ and $\displaystyle e.x+f$ with

$\displaystyle c,\ d,\ e,\ f$ all natural numbers. Then:

$\displaystyle x^2+a.x+b = (c.x+d)(e.x+f)$

$\displaystyle =c.e.x^2+(d.e+c.f).x+d.f$

.

Now equate coefficients of corresponding powers of x:

$\displaystyle c.e=1$

$\displaystyle d.e+c.f=a$

$\displaystyle d.f=b$

As $\displaystyle c$ and $\displaystyle e$ are natural numbers

the first of these equations implies $\displaystyle c=e=1$.

Substituting these into the second equation gives:

$\displaystyle d+f=a$

Now the third of these equations implies both $\displaystyle d$ and

$\displaystyle f$ are odd as by assumption $\displaystyle b$ is odd.

But $\displaystyle d$ and $\displaystyle f$ odd implies that their sum is

even, but by assumption $\displaystyle a$ is odd and is equal to the sum

of $\displaystyle d$ and $\displaystyle f$ which is even - a contradiction.

Hence $\displaystyle x^2+a.x+b$ with $\displaystyle a$ and $\displaystyle b$

odd natural numbres, cannot be factored into a product

of two such linear factors.

RonL